A molecule is linear without a center of inversion. What is its point group?
The first thing we need to address is what makes a molecule linear. When we describe a molecule’s shape, we describe the configuration of the center of each atom or ion. We class the molecule as linear if a single straight line can be drawn through all the center points. A center of inversion is a symmetry element through which the operation of inversion can occur, where every 𝑥-, 𝑦-, 𝑧-coordinate is moved through to the other side.
For a symmetrical diatomic like oxygen, the center of inversion lies in the middle of the bond between the two particles. If we move each particle center through the center of inversion the same distance out the other side, it produces exactly the same molecule. In this question, our molecule does not have a center of inversion. A diatomic molecule where the two atoms are not equivalent does not have a center of inversion because, even though we could perform the operation, we would not produce an identical molecule.
So, we have a molecule that’s linear and without a center of inversion. One example of such a molecule is hydrogen chloride HCL. So, going back to our three-atom linear example, we would need the atoms on the outside to be different. The question doesn’t give us any more details about the identity of the chemical. But we’re still asked to identify its point group.
A point group is a summary of symmetry elements. So, a molecule with a point group has all those symmetry elements and no others. To find the point group of a molecule, we need either all the symmetry elements, or we need to find the flow chart to the correct point group.
The first question on the point group flow chart is whether the molecule is linear or not. Obviously, for this question, the answer is yes. The next question on our flow chart is whether the molecule has a center of inversion, given the symbol 𝑖. The answer to this question is no. On our point group flow chart, this leads us to the answer 𝐶∞𝑣.
The 𝐶∞𝑣 point group has the 𝐸 symmetry element, whose operation leaves the molecule unchanged, a 𝐶∞ proper axis of rotation, and an infinite number of mirror planes align vertically with the principal rotation axis. These are the only symmetry elements possessed by a linear molecule without a center of inversion. With a center of inversion, the point group would be 𝐷∞𝑣. So, the answer to our question, what’s the point group of a linear molecule without a center of inversion, is 𝐶∞𝑣.